Quantum Fokker-Planck models: the Lindblad and Wigner approaches

In this article we try to bridge the gap between the quantum dynamical semigroup and Wigner function approaches to quantum open systems. In particular we study stationary states and the long time asymptotics for the quantum Fokker-Planck equation. Our new results apply to open quantum systems in a harmonic confinement potential, perturbed by a (large) sub-quadratic term.Comment: 19 pages, corrected typos and quoted Theorem 6 more precisel

Large-time behavior in non-symmetric Fokker-Planck equations

We consider three classes of linear non-symmetric Fokker-Planck equations having a unique steady state and establish exponential convergence of solutions towards the steady state with explicit (estimates of) decay rates. First, "hypocoercive" Fokker-Planck equations are degenerate parabolic equations such that the entropy method to study large-time behavior of solutions has to be modified. We review a recent modified entropy method (for non-symmetric Fokker-Planck equations with drift terms that are linear in the position variable). Second, kinetic Fokker-Planck equations with non-quadratic potentials are another example of non-symmetric Fokker-Planck equations. Their drift term is nonlinear in the position variable. In case of potentials with bounded second-order derivatives, the modified entropy method allows to prove exponential convergence of solutions to the steady state. In this application of the modified entropy method symmetric positive definite matrices solving a matrix inequality are needed. We determine all such matrices achieving the optimal decay rate in the modified entropy method. In this way we prove the optimality of previous results. Third, we discuss the spectral properties of Fokker-Planck operators perturbed with convolution operators. For the corresponding Fokker-Planck equation we show existence and uniqueness of a stationary solution. Then, exponential convergence of all solutions towards the stationary solution is proven with an uniform rate

On multi-dimensional hypocoercive BGK models

We study hypocoercivity for a class of linearized BGK models for continuous phase spaces. We develop methods for constructing entropy functionals that enable us to prove exponential relaxation to equilibrium with explicit and physically meaningful rates. In fact, we not only estimate the exponential rate, but also the second time scale governing the time one must wait before one begins to see the exponential relaxation in the L1 distance. This waiting time phenomenon, with a long plateau before the exponential decay "kicks in" when starting from initial data that is well-concentrated in phase space, is familiar from work of Aldous and Diaconis on Markov chains, but is new in our continuous phase space setting. Our strategies are based on the entropy and spectral methods, and we introduce a new "index of hypocoercivity" that is relevant to models of our type involving jump processes and not only diffusion. At the heart of our method is a decomposition technique that allows us to adapt Lyapunov's direct method to our continuous phase space setting in order to construct our entropy functionals. These are used to obtain precise information on linearized BGK models. Finally, we also prove local asymptotic stability of a nonlinear BGK model.Comment: 55 pages, 2 figure